In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). A theory of pseudogroups was developed by Élie Cartan in the early 1900s.^[1][2] Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This picture illustrates how the hours on a clock form a group under modular addition. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

It is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E. The groupoid condition on those is fulfilled, in that homeomorphisms In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...

h:U → V

and

g:V → W

compose to a homeomorphism from U to W. The further requirement on a pseudogroup is related to the possibility of patching (in the sense of descent, transition functions, or a gluing axiom). If we are given instead In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ... In mathematics, a transition function has several different meanings: In topology, a transition function is a homeomorphism from one coordinate chart to another. ... In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor F: Open(X) → C to a category C which initially one takes to be the category...

g′:V′ → W

and define V* to be the intersection of V and V′, we should be able to compose restricted functions h* and g′*. Here In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In mathematics, a domain of a k-place relation L ⊆ X1 × … × Xk is one of the sets Xj, 1 ≤ j ≤ k. ...

h*:U* → V*

is the restriction to U*, defined as h⁻¹(V*); and W* is g(V*).

An example in space of two dimensions is the pseudogroup of invertible holomorphic functions of a complex variable (invertible in the sense of having an inverse function). The properties of this pseudogroup are what makes it possible to define Riemann surfaces by local data patched together. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

In general, pseudogroups were studied as a possible theory of infinite-dimensional Lie groups. The concept of a local Lie group, namely a pseudogroup of functions defined in neighbourhoods of the origin of E, is actually closer to Lie's original concept of Lie group, in the case where the transformations involved depend on a finite number of parameters, than the contemporary definition via manifolds. One of Cartan's achievements was to clarify the points involved, including the point that a local Lie group always gives rise to a global group, in the current sense (an analogue of Lie's third theorem, on Lie algebras determining a group). The formal group is yet another approach to the specification of Lie groups, infinitesimally. It is known, however, that local topological groups do not necessarily have global counterparts. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... The factual accuracy of this article is disputed. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... This article does not cite any references or sources. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a formal group law is (roughly speaking) the formal power series analogue of a Lie group. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...

Examples of infinite-dimensional pseudogroups abound, beginning with the pseudogroup of all diffeomorphisms of E. The interest is mainly in sub-pseudogroups of the diffeomorphisms, and therefore with objects that have a Lie algebra analogue of vector fields. Methods proposed by Lie and by Cartan for studying these objects have become more practical given the progress of computer algebra. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...

In the 1950s Cartan's theory was reformulated by Shiing-Shen Chern, and a general deformation theory for pseudogroups was developed by Kunihiko Kodaira and D. C. Spencer. In the 1960s homological algebra was applied to the basic PDE questions involved, of over-determination; this though revealed that the algebra of the theory is potentially very heavy. In the same decade the interest for theoretical physics of infinite-dimensional Lie theory appeared for the first time, in the shape of current algebra. Chen Xingshen Shiing-Shen Chern (陳省身; pinyin: Chén XǐngshÄ“n; October 26, 1911 – December 3, 2004) was a Chinese-American mathematician, one of the leading differential geometers of the twentieth century. ... In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. ... Kunihiko Kodaira (小平 邦彦 Kodaira Kunihiko, 16 March 1915 – 26 July 1997) was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds; and as the founder of the Japanese school of algebraic geometers. ... Donald C. Spencer (April 25, 1912 - December 23, 2001) was an American mathematician, known for major work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of partial differential equations. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... This does not cite its references or sources. ...

References

1. ^ Cartan, Élie (1904). "Sur la structure des groupes infinis de transformations". Annales scientifiques de l'É.N.S. 21: 153-206. 
2. ^ Cartan, Élie (1909). "Les groupes de transformations continus, infinis, simples". Annales scientifiques de l'École Normale Supérieure Sér. 3 26: 93-161. 

External links

• Alekseevskii, D.V., Pseudo-groups, SpringerLink Encyclopaedia of Mathematics (2001)